I'm going to address your question "is this covered under ADA?"
For certain, schizoaffective disorder is a qualifying condition under the ADA. To be a qualifying condition, it must be a physical or mental impairment that substantially limits a major life activity. There's no question it qualifies. Your employer will also be covered -- every hospital is large enough to be covered by the ADA.
But now the question is what accommodation will they be willing to offer you. The ADA requires that your employer make reasonable accommodations as long as they do not interfere with the essential functions of the position. So the first question will be whether working nights is an essential function. It's possible that the program may list "working shifts of various lengths and times including night shifts" as an essential function. if so, then they don't need to compromise on it at all.
If they have not listed it as an essential function, then the process of coming to a reasonable accommodation starts. And this is where things get really tricky, since the gov't and courts have not defined what "reasonable" is. The only definition is "looking at court cases and interpreting what the courts think". This is really messy, since no two cases are identical and courts have made incompatible decisions.
You don't just get to define what the accommodation is -- your employer is able to discuss and review it with you. If your problem was narcolepsy (for example), instead of having you do no night shifts they could have you wear a device that senses if you're falling asleep and wakes you up.
In your situation that's not a reasonable solution. What a program might do decide to schedule your shifts differently. For example, let's assume that the day shifts run 730A to 530P, and night shifts 530P to 730A. Your problem is missing sleep, so the program could reasonably infer that you could work from 530P to midnight. Since that's about half the length of a full night shift, they could insist that you work twice as many of those instead of your night shifts. However, you can't work until midnight and then start at 730A - not enough sleep, and also a duty hour violation (need an 8 hour break). So what a program might do is have you work on Friday and Sat nights when you're on elective blocks. A decent IM program might have 3-4 months of outpatient / elective blocks in the PGY-1, and 4-6 weeks of nights. Five nights / week times five weeks of nights is 25 night shifts -- assuming you worked an evening shift for each of those, you'd "owe" another 25 evening shifts. 3 months = 12 weeks of electives -- you might end up working every weekend.
The next possible problem is curriculum. The program could claim (reasonably) that they are providing both employment and an education, and working night shifts is part of the curriculum. They could try to decline to hire you on this basis, or perhaps decide to not give you full credit by the ABIM since you didn't complete the whole curriculum. (This seems unlikely to me)
And then there's getting licensed. Some states may not give you a medical license, especially if you have a history of psychotic episodes / hallucinations. If you don't get a license, you lose your spot since you're no longer qualified.
You're left with a tough choice. By law, you don't need to disclose this at all when you're applying. When you apply you're asked whether you can do the job with or without reasonable accommodations, and you simply answer "yes" since "no nights" is a reasonable accommodation as far as you're concerned. Then, once you have a spot, you disclose your disability and start working with your new employer as to how they are going to accommodate it. You don't know if they'll say "it's fine, don't worry about it" or if they will come up with some draconian accommodation. If they try to not accommodate you at all, or come up with something you can't work with, then you're stuck either resigning or taking them to court - neither a great option.
Or, you could optionally disclose the issue. This will absolutely, positively decrease the number of interviews you get, or if you disclose after the interview likely impact your ranking (although that would be illegal discrimination). So your options and chances of matching would be lower. But if you did match, you would know that the program was willing to work with you.
In your case, schizoaffactive disorder has a very bad connotation to it. If you disclose it, especially as an IMG, I expect you will get nothing. So in this case, your likely only option is to not mention it at all and then hope for the best. If you can find programs with no nights, then those are obviously worth targeting -- although they may be quite competitive for that reason. It almost certainly means you'll be starting off with an adversarial relationship with your program, which means if you start having problems they are much less likely to cut you any slack.
Best of luck
